Global dynamics and diffusion-driven pattern formation in a predator-prey system with two chemicals

TL;DR

This paper investigates a complex predator-prey model with two chemicals, establishing conditions for solution existence, stability, and pattern formation through numerical Turing bifurcation analysis.

Contribution

It provides the first numerical criteria for Turing bifurcation in a four-equation predator-prey-chemical system, highlighting predation's role in pattern formation.

Findings

Global existence of solutions under certain conditions

Asymptotic stability of homogeneous steady states

Numerical Turing bifurcation diagrams showing pattern emergence

Abstract

This work analyzes a predator-prey cross-diffusion system coupled with two chemical substances under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R^n (n >= 2) with smooth boundary dOmega. Under appropriate conditions on the model parameters, the global existence of classical solutions is established. Furthermore, by constructing a suitable Lyapunov functional, the asymptotic stability of the spatially homogeneous steady state is proved. The emergence of spatial patterns induced by diffusion-driven instability is also investigated. Owing to the complexity of the resulting four-equation system, the criteria for Turing bifurcation are derived numerically rather than analytically. Numerical simulations are performed to generate Turing bifurcation diagrams, illustrating the dynamical responses of the system to variations in the predation rate. These results provide new insights into the role of predation intensity in the formation of spatial patterns in predator-prey systems mediated by two chemical substances.